Many quantum effects and quantum technologies rely on fragile quantum fluctuations that can easily be suppressed by thermal fluctuations. It is therefore critical to overcome thermal fluctuations for observing these quantum effects. This is why most quantum technologies and applications require cooling or alternative ways of suppressing thermal fluctuations. Often these techniques require sophisticated apparatuses, e.g. cooling in dilution fridge. There are however algorithmic techniques for cooling which are less demanding from this point of view.
Heat-bath algorithmic cooling (HBAC) is a proposed method for decreasing entropy in a quantum system, resulting in an improvement in the purity of quantum states of the quantum system.
HBAC operates on an ensemble of qubits and effectively cools down and purifies a subset of the qubits in the ensemble. HBAC drives the system out of equilibrium by transferring the entropy from target qubits to the rest of the ensemble, which may be referred to as refrigeration qubits. The target qubits may also be referred to as the “computation qubits” and the refrigerant qubits are referred to as the “reset qubits”. HBAC was first introduced for a closed system using compression algorithms. For closed system HBAC, the cooling is limited by the Shannon bound for compression. It was later proposed to use a heat-bath to enhance the cooling beyond the Shannon bound.
The achievable purity utilizing proposed HBAC techniques is physically limited and the limit can be achieved only symptomatically. The most optimal HBAC technique that has been previously proposed is known as the partner pairing algorithm (PPA). The asymptotic state of PPA, which reaches the cooling limit, may be referred to as the optimal asymptotic cooling state (OAS).
Although PPA is the optimal technique for HBAC, in practice it is too complex and is not suitable for experimental purposes. One of the main challenges of PPA is that it requires sorting the diagonal of the density matrix in each iteration. These sort operations depend on the state and, because the state changes through the process, the unitary operation for implementing the sort would change as well. In PPA, the sort operation is not a fixed unitary gate and for each iteration, classical computation is required to find the unitary operator that implements the sort for that specific iteration. Computing the unitary operator must be repeated for every iteration, which is computationally taxing. Also, the experimental control would need to change for each iteration in order to implement the specific unitary operation required.
Therefore, PPA is a time and state-dependent process, which leads to several critical problems for “practical” applications of this technique. Here we use “practical” for an algorithm if its process is not time-dependent and is robust to deviations from the expected state in each iteration, and therefore is more experimentally feasible than algorithms that are, for example, time-dependent.
In theory, all the operations in PPA may be pre-computed. However, in practice, small imperfections change the state of the system and the pre-computed operators cannot not sort the diagonal elements of the perturbed density matrix. In order to account for the imperfections, techniques like quantum state tomography would be required to monitor the state of the system, which is not practical in real world applications because, experimentally, tomography is not perfect and involves some estimation errors. Further, even if monitoring the system were possible, the imperfections may affect the result of the PPA, possibly resulting in heating the target qubits and may even not converge.
The ideal HBAC technique should be robust and practically simple and at the same time, converge to the OAS.
Improvements in decreasing entropy in quantum systems are desirable.